Constrain vertex positions to moving objects?
You can constrain each individual vertices of your polygon to the joints of your mechanism instead of moving the mesh origin. No need to move the mesh origin.

You can do it using the Hook modifier. Create one Vertex group for each vertices of your polygon, then, using the Hook modifier, constrain each of those group to one of the joint of your mechanism, using whatever mesh that makes up the joint that you think would show the best result.

Alternatively, you can use some Empty, constrain them to the center of the joints, then Hook each Vertex group to one of the Empty.